Icositrigon: Difference between revisions

A regular icositrigon has [[internal angle]]s of <math display="inline">\frac{3780}{23}</math> degrees, with an area of <math display="inline">A = \frac{23}{4}a^2 \cot \frac{\pi}{23} = 23r^2 \tan \frac{\pi}{23} \simeq 41.8344\,a^2,</math> where <math>a</math> is side length and <math>r</math> is the inradius, or [[apothem]].

InThe additionregular toicositrigon notis beingnot [[Constructible polygon|constructible]] with a [[Straightedge and compass construction|compass and straightedge]] or [[angle trisection]],<ref>''{{Cite OEIS|A048136|Tomahawk-nonconstructible <math>n</math>-gons'' [[OEIS]]; https://oeis.org/A048136}}</ref> on account of the [[23 (number)|number 23]] being neither a [[Fermat prime|Fermat]] nor [[Pierpont prime#Polygon construction|Pierpont prime]]. In addition, the regular icositrigon is the [[Neusis construction#Use of the neusis|smallest regular polygon that is not constructible even with neusis]], after the discovery of neusis construction of the [[hendecagon]] by Elliot Benjamin and Chip Snyder in 2014.<ref>Benjamin, Elliot; Snyder, C. Mathematical Proceedings of the Cambridge Philosophical Society156.3 (May 2014): 409-424.; https://dx.doi.org/10.1017/S0305004113000753</ref>

Suppose <math>\alpha</math> in <math>\CComplex</math> is constructible using a compass and twice-notched

for which the index [<math>[K_j</math> : K_{{math|''K''<sub>''j−1''j - 1}]</submath>}}] at each step is 2, 3, 5, or 6. In particular, if <math>N =</math> [<math>K</math> :

<math>\Q]</math>], then the only primes dividing <math>N</math> are 2, 3, and 5. (Theorem 5.1)

If we can construct the regular p-gon, then we can construct <math>\zeta_p = e^\frac{2\pi i}{p}</math>, which is the root of an [[irreducible polynomial]] of degree <math>p</math> {{math|−}}- <math>1</math>. By Theorem 5.1, <math>\zeta_p</math> lies in a field <math>K</math> of degree <math>N</math> over <math>\Q</math>, where the only primes that divide <math>N</math> are 2, 3, and 5. But <math>\Q</math>[<math>\zeta_p]</math>] is a subfield of <math>K</math>, so <math>p</math> {{math|−}}- <math>1</math> divides <math>N</math>. In particular, for <math>p = 23</math>, <math>N</math> must be divisible by 11, and for <math>p = 29</math>, ''N'' must be divisible by 7.<ref>Arthur{{cite journal
| last1=Baragar (| first1=Arthur
| date=2002)
| title=Constructions Using a Compass and TwiceNotchedTwice-Notched Straightedge,
| journal=The American Mathematical Monthly, 109:2, 151-164, DOI:

Below is a table of fiveten regular icositrigrams, or [[Star polygons|star]] 23-gons, labeled with their respective [[Schläfli symbol]] {23/q}, 2 <small><math>\leq</math></small>≤ q <small><math>\leq</math></small>≤ 11 where q is prime.

==External Linkslinks==